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Kernel based data fusion Discussion of a Paper by G. Lanckriet 1 Paper 2 Overview Problem: Aggregation of heterogeneous data Idea: Different data are represented by different kernels Question: How to combine different kernels in an elegant/efficient way? Solution: Linear combination and SDP Application: Recognition of ribosomal and membrane proteins 3 Linear combination of kernels xTKx = x2K x2 K weight kernel 0 x Resulting kernel K is positive definite (xTKx > 0 for x, provided i > 0 and xTKi x > 0 ) Elegant aggregation of heterogeneous data More efficient than training of individual SVMs KCCA uses unweighted sum over individual kernels 4 Support Vector Machine square norm vector penalty term Hyperplane slack variables 5 Dual form quadratic, convex scalar 0 positive definite Lagrange multipliers Maximization instead of minimization Equality constraints Lagrange multipliers instead of w,b, Quadratic program (QP) 6 Inserting linear combination ugly Fixed trace, avoids trivial solution Combined kernel must be within the cone of positive semidefinite matrices 7 Cone and other stuff Positive semidefinite cone: A Positive semidefinite: xTAx ≥ 0, x The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone. http://www.convexoptimization.com/dattorro/positive_semidefinate_cone.html 8 Semidefinite program (SDP) Fixed trace, avoids trivial solution positive semidefinite constraints 9 Dual form quadratic constraint Quadratically constraint quadratic program (QCQP) QCQPs can be solved more efficiently than SDPs (O(n3) <-> O(n4.5)) Interior point methods 10 Interior point algorithm Linear program: maximize cTx subject to Ax < b x≥0 Classical Simplex method follows edges of polyhedron Interior point methods walk through the interior of the feasible region 11 Application Recognition of ribosomal and membrane proteins in yeast 3 Types of data Amino acid sequences Protein protein interactions mRNA expression profiles 7 Kernels Empirical kernel map -> sequence homology FFT -> sequence hydropathy BLAST(B), Smith-Waterman(SW), Pfam KD hydropathy profiles, padding, low-pass filter, FFT, RBF Interaction kernel(LI) -> PPI Diffusion(D) -> PPI RBF(E) -> gene expression 12 Results Combination of kernels performs better than individual kernels Gene expression (E) most important for ribosomal protein recognition PPI (D) most important for membrane protein recognition 13 Results Small improvement compared to weights = 1 SDP robust in the presence of noise How performs SDP versus kernel weights derived from accuracy of individual SVMs? Membrane protein recognition Other methods use sequence information only TMHMM designed for topology prediction TMHMM not trained on yeast only 14 Why is this cool? Everything you ever dreamed of: Optimization of C included (2-norm soft margin SVM =1/C) Hyperkernels (optimize the kernel itself) Transduction (learn from labeled & unlabeled samples in polynomial time) SDP has many applications (Graph theory, combinatorial optimization, …) 15 Literature Learning the kernel matrix with semidefinite programming G.R.G.Lanckrit et. al, 2004 Kernel-based data fusion and its application to protein function prediction in yeast G.R.G.Lanckrit et. al, 2004 Machine learning using Hyperkernels C.S.Ong, A.J.Smola, 2003 Semidefinite optimization M.J.Todd, 2001 http://www-user.tu-chemnitz.de/~helmberg/semidef.html 16 Software SeDuMi (SDP) Mosek (QCQP, Java,C++, commercial) YALMIP (Matlab) … http://www-user.tu-chemnitz.de/~helmberg/semidef.html 17